Table Of ContentT H E F R O N T I E R S C O L L E C T I O N
Wouter Schmitz
U NDERSTANDING
R ELATIVITY
A Conceptual Journey
Into Spacetime, Black Holes and
Gravitational Waves
123
The Frontiers Collection
Series Editors
Avshalom C. Elitzur, Iyar, Israel Institute of Advanced Research, Rehovot, Israel
Zeeya Merali, Foundational Questions Institute, Decatur, GA, USA
Maximilian Schlosshauer, Department of Physics, University of Portland, Portland,
OR, USA
Mark P. Silverman, Department of Physics, Trinity College, Hartford, CT, USA
Jack A. Tuszynski, Department of Physics, University of Alberta, Edmonton, AB,
Canada
Rüdiger Vaas, Redaktion Astronomie, Physik, bild der wissenschaft,
Leinfelden-Echterdingen, Germany
The books in this collection are devoted to challenging and open problems at the
forefront of modern science and scholarship, including related philosophical debates.
In contrast to typical research monographs, however, they strive to present their topics
in a manner accessible also to scientifically literate non-specialists wishing to gain
insight into the deeper implications and fascinating questions involved. Taken as a
whole, the series reflects the need for a fundamental and interdisciplinary approach
to modern science and research. Furthermore, it is intended to encourage active
academics in all fields to ponder over important and perhaps controversial issues
beyond their own speciality. Extending from quantum physics and relativity to
entropy, consciousness, language and complex systems—the Frontiers Collection
will inspire readers to push back the frontiers of their own knowledge.
Wouter Schmitz
Understanding Relativity
A Conceptual Journey Into Spacetime, Black
Holes and Gravitational Waves
Wouter Schmitz
Amsterdam, Noord-Holland
The Netherlands
ISSN 1612-3018 ISSN 2197-6619 (electronic)
The Frontiers Collection
ISBN 978-3-031-17218-2 ISBN 978-3-031-17219-9 (eBook)
https://doi.org/10.1007/978-3-031-17219-9
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
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To all curious minds
It is worth taking space and time to be curious
For it brings us the beauty of understanding
And there is so much beauty in understanding more
Of the inner workings of spacetime
Hold on to that beauty!
Keep it from getting forever lost
In the black hole of stress
Which eats our space and time away
Leaving us small and beauty unreachable
Beyond the horizon
Wouter Schmitz, 2022
Contents
1 Introduction .................................................. 1
2 What is (the) Matter? .......................................... 7
2.1 The Postulates of Special Relativity ......................... 7
2.2 Why We Are Waves ...................................... 9
2.3 Wave Dynamics .......................................... 12
3 On the Velocity of Massless Waves .............................. 17
3.1 Maxwell’s Speed of Light ................................. 17
3.2 Generic Speed of Light .................................... 19
3.3 Elasticity of the Vacuum .................................. 20
3.4 The Problem of the Ether .................................. 21
4 On the Velocity of Massive Waves ............................... 25
4.1 Wave Velocity ........................................... 25
4.2 A Game with Rope and Springs ............................ 26
4.3 Consequence 1: You Cannot Go Faster Than Light ............ 28
4.4 Consequence 2: The Relation Between Frequency
and Wavelength Depends on the Mass ....................... 28
4.5 Consequence 3: Mass = Inertia ............................. 32
4.6 Consequence 4: Other Potential Differences Can Create
“Mass” ................................................. 35
4.7 Consequence 5: Mass Can Be Changed into Energy ........... 37
4.8 Example: Photons in a Plasma ............................. 38
4.9 What These Consequences Say ............................. 38
4.10 Length Contraction and Time Dilation of Waves .............. 40
4.11 Doppler Effect ........................................... 49
4.12 Conclusions Regarding the Way We Perceive Distance
in Spacetime ............................................. 53
4.13 The Postulates of Special Relativity ......................... 55
4.14 About Higgs ............................................. 56
vii
viii Contents
5 The Relative, the Absolute, and the Paradoxical .................. 59
5.1 Why Would a Distance in Space Care About Our Waves? ...... 59
5.2 The Clash of the Observers ................................ 61
5.3 The Twin Paradox ........................................ 62
5.4 Einstein’s Train .......................................... 69
5.5 Extreme Energies and the Continuity of Spacetime ............ 70
5.6 Conclusion .............................................. 72
6 The Minkowski Metric ........................................ 73
6.1 Defining the Metric ....................................... 73
6.2 Origin of the Minkowski Metric ............................ 75
6.3 Coordinates in Minkowski Spacetime ....................... 78
6.4 Simultaneity ............................................. 81
6.5 The Light Cone .......................................... 84
6.6 Causality of Spacetime .................................... 90
6.7 Boosts .................................................. 94
6.8 Interpretations of the Minkowski Metric ..................... 96
6.9 The Debate: Contracting Waves Versus Contracting Space ..... 99
6.10 Bell’s Paradox of the Accelerating Spaceships ................ 101
7 The Equivalence Principle ..................................... 105
7.1 Two Different Meanings of “Mass” ......................... 105
7.2 Inertial Mass = Gravitational Mass ......................... 107
7.3 The Equivalence Principle ................................. 109
7.4 Can Gravity Bend a Light Beam? ........................... 114
7.5 How Does Mass Curve Spacetime? ......................... 116
7.6 Speculating About the Structure of the Vacuum ............... 119
7.7 Inertia and How Spacetime Tells Mass How to Move .......... 120
7.8 Perception or Reality? .................................... 127
7.9 Conclusion .............................................. 128
8 The Curvature of Spacetime .................................... 131
8.1 What is a Curved Space? .................................. 131
8.2 Riemannian Manifolds .................................... 133
8.3 Making Curved Space from a Series of Flat Spaces ............ 135
8.4 How to Curve Spacetime .................................. 139
8.5 Energy Conservation and Gravity ........................... 147
8.6 Geodesics ............................................... 148
8.7 Tidal Forces ............................................. 150
9 How Waves Change Shapes and Volumes ........................ 153
9.1 Changing Volumes and Changing Shapes .................... 153
9.2 Waves as a Source of Gravitation ........................... 163
9.3 Conceptual Understanding of the Einstein Equation ........... 167
9.4 Schwarzschild Metric ..................................... 172
Contents ix
10 The Effects of Curvature ....................................... 185
10.1 The GPS System ......................................... 185
10.2 Gravitational Redshift and Doppler Effect ................... 186
10.3 Geodesics Revisited ...................................... 187
10.4 Mercury’s Orbit .......................................... 196
10.5 Orbits of Light ........................................... 202
10.6 Geodetic Precession ...................................... 207
10.7 Frame Dragging .......................................... 213
11 A Journey to the Centre of a Black Hole ......................... 221
11.1 Mass Limits and Collapsing Stars ........................... 221
11.2 Pinching the Light Cone ................................... 227
11.3 The Singularity .......................................... 235
11.4 Hawking Radiation and the Unruh Effect .................... 239
11.5 Thermodynamics and Entropy of a Black Hole ............... 248
11.6 Rotating Black Holes ..................................... 252
11.7 Charged Black Holes ..................................... 262
12 Black Hole Phenomena ........................................ 265
12.1 Gravitomagnetism and Jets ................................ 265
12.2 Analysing the Photos of the Black Hole in M87 ............... 275
12.3 The Black Hole in Our Own Galaxy ........................ 282
12.4 Black Hole Simulations in Analogue Physics ................. 284
12.5 Wormholes and White Holes ............................... 286
13 Gravitational Waves ........................................... 303
13.1 Symmetry in Spacetime ................................... 305
13.2 The Production and Propagation of Gravitational Waves ....... 308
13.3 The Effect of Gravitational Waves .......................... 312
13.4 Observations and a Further Test of Einstein’s General
Relativity ............................................... 318
13.5 What Gravitational Waves Tell Us About the Nature
of Spacetime ............................................ 324
13.6 The Measurement Approach and Detectors ................... 325
14 All that Matters in the Dark Universe ........................... 333
14.1 The Expanding Universe .................................. 333
14.2 Observations and Characteristics of Dark Matter .............. 351
14.3 Dark Energy and the Cosmological Model ................... 361
14.4 Observations and the Hubble Controversy ................... 368
14.5 The Big Bang ............................................ 381
15 Emergent Gravity and Conclusion .............................. 393
15.1 The Emergence of Spacetime and Gravity from Quantum
States ................................................... 394
15.2 Conclusions ............................................. 402
x Contents
References ........................................................ 405
Index ............................................................. 411
